Properties

Degree 2
Conductor 31
Sign $0.989 - 0.141i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.557 + 0.405i)2-s + (−0.824 − 0.367i)3-s + (−0.471 − 1.44i)4-s + (−1.85 + 3.21i)5-s + (−0.311 − 0.538i)6-s + (0.510 − 0.567i)7-s + (0.750 − 2.31i)8-s + (−1.46 − 1.62i)9-s + (−2.33 + 1.03i)10-s + (4.02 + 0.855i)11-s + (−0.143 + 1.36i)12-s + (0.304 + 2.89i)13-s + (0.514 − 0.109i)14-s + (2.70 − 1.96i)15-s + (−1.11 + 0.807i)16-s + (−1.27 + 0.272i)17-s + ⋯
L(s)  = 1  + (0.394 + 0.286i)2-s + (−0.475 − 0.211i)3-s + (−0.235 − 0.724i)4-s + (−0.829 + 1.43i)5-s + (−0.127 − 0.219i)6-s + (0.193 − 0.214i)7-s + (0.265 − 0.817i)8-s + (−0.487 − 0.541i)9-s + (−0.738 + 0.328i)10-s + (1.21 + 0.257i)11-s + (−0.0415 + 0.394i)12-s + (0.0843 + 0.802i)13-s + (0.137 − 0.0292i)14-s + (0.698 − 0.507i)15-s + (−0.277 + 0.201i)16-s + (−0.310 + 0.0659i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.989 - 0.141i$
motivic weight  =  \(1\)
character  :  $\chi_{31} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 31,\ (\ :1/2),\ 0.989 - 0.141i)$
$L(1)$  $\approx$  $0.680268 + 0.0483898i$
$L(\frac12)$  $\approx$  $0.680268 + 0.0483898i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 31$, \(F_p\) is a polynomial of degree 2. If $p = 31$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad31 \( 1 + (1.63 + 5.32i)T \)
good2 \( 1 + (-0.557 - 0.405i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.824 + 0.367i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (1.85 - 3.21i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.510 + 0.567i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (-4.02 - 0.855i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.304 - 2.89i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (1.27 - 0.272i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.397 + 3.77i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (1.01 - 3.12i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.96 + 2.87i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (-5.20 - 9.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.690 + 0.307i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (-0.748 + 7.11i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (-0.708 + 0.514i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.39 - 2.65i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (-0.847 - 0.377i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + (-1.04 + 1.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.17 + 5.74i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (-5.53 - 1.17i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (-13.7 + 2.93i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (12.8 - 5.73i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (1.37 + 4.21i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (1.63 + 5.01i)T + (-78.4 + 57.0i)T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.06848893303481188828549091183, −15.36316272134762891054513202424, −14.73653459317990643314113371739, −13.75641095488656317955596332716, −11.74588415655120802281211696997, −11.06366485160697674112665060806, −9.418449148627935586510313399827, −7.09706963470709097559592763467, −6.24146006614560584595613256991, −4.02894472949931429174626488087, 4.01676555322226381565934810542, 5.32364987605313280391396869349, 8.003119174704505119944117621173, 8.889321713410093184860549490613, 11.18779269540851023814194574804, 12.10140728392505623702518237012, 12.91816189161627770017170533013, 14.41533391248187578855309062397, 16.26111349419193082015783042272, 16.66920327922827264686377880385

Graph of the $Z$-function along the critical line